Category Theory For Beginners: Understanding Pullbacks
Pullbacks are simple structures that can encode many important ideas in category theory. We describe many results about pullbacks, and how they dualize to pushouts. In particular, we consider how pullbacks can describe ideas in real world language. We describe how products, equalizers, and monica can be described in terms of pullbacks. We describe how inverse images, and the intersection of subobjects can be described in terms of monica. We also mention the pullback lemma, and how pullbacks can be described in terms of products and equalizers.
The latter result, and more theory is covered in these two unlisted videos:
• Subobjects and pullbacks 1
• Subobjects and pullbacks 2
A proper statement and proof of the pullback lemma(s) can be found in the following videos:
• Pullback Lemma 1
• Pullback Lemma 2
• Pullback Lemma 3
• Pullback Lemma 4 example
• Pullback Lemma 5
• Pullback Lemma 6
A full description of how to find general limits and colimits (of which pullbacks, and pushouts are respectively examples) in categories of functors into Set can be found here:
• Category Theory For Beginners: Preshe...
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